Kato Yuki

Abstract. As Nakayama’s lemma, in commutative ring theory, it is well-known that the residue functor of an ideal contained in the Jacobson radical is conservative. In almost mathematics setting, considering an almost version of perfect complexes, we prove an almost derived analogue of Nakayama’s lemma, stating the conservativity of the derived functor of the residue of a tight ideal contained in the Jacobson radical.

This note is a revisiting Quillen's note [3] "Modules theory over non-unital rings" in category theory language. Faltings [1]; Gabber and Ramero [2] established almost mathematics which is the same as Quillen's bilocalization of a category of modules by almost zero objects.